There are 27 players in a ping-pong tournament. each player plays every other player exactly once. how many games will be played?

If x people play each other in pairs exactly once, then there will be
hmm

if 1, then 0 games
if 2, then 1 game
if 3 then 3 games ab, ac, bc
if 4 then ab, ac, ad, bc, bd, cd then 6 games
if 5 then ab, ac, ad, ae, bc, bd, be, cd, ce, de
we see a pattern
summation
first term is 0, when n=1
increases by 1
so
simlified to


for n games, a total of games are played

for 27 players, there are a total of 27(13)=351 games

Agc and bgd are vertical angles